Mass shell smearing effects in top pair production
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International Journal of Modern Physics Ac (cid:13)
World Scientific Publishing Company
MASS SHELL SMEARING EFFECTS IN TOP PAIR PRODUCTION
V. I. KUKSA ∗ Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia
R. S. PASECHNIK † Theoretical High Energy Physics, Department of Astronomy and Theoretical Physics,Lund University, SE 223-62 Lund, Sweden
D. E. VLASENKO ‡ Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia
Received Day Month YearRevised Day Month YearThe top quark pair production and decay are considered in the framework of the smeared-mass unstable particles model. The results for total and differential cross sections invicinity of t ¯ t threshold are in good agreement with the previous ones in the literature.The strategy of calculations of the higher order corrections in the framework of themodel is discussed. Suggested approach significantly simplifies calculations compared tothe standard perturbative one and can serve as a convenient tool for fast and precisepreliminary analysis of processes involving intermediate time-like top quark exchangesin the near-threshold region. Keywords : top quark; top pair production; unstable particlesPACS number: 11.10.St
1. Introduction
The top pair production and decay are the key processes for precision tests of theStandard Model (SM) (see e.g. Ref. [1] and references therein). They were intensivelystudied in the framework of the Quantum Chromodynamics (QCD) and Electro-Weak (EW) perturbation theory during last two decades, and various methods andschemes were proposed. The major goal of these investigations is to define the basicphysical parameters of the top quark, such as its mass, width and couplings withother SM particles. In the past, the top quark physics was one of the primaryresearch objectives at Tevatron. Nowadays, the biggest attention is paid to the ∗ [email protected] † [email protected] ‡ [email protected] 1 eptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final V. I. Kuksa, R. S. Pasechnik, D. E. Vlasenko process of the top quark production at the LHC (see e.g. Refs. [2, 3]). However, thehighest precision measurements of the top quark properties can best be reached atthe future Linear Collider (LC) which supposedly operates in a clean experimentalenvironment. The top quark physics is one of the most interesting and challengingtargets for future e + e − or µ + µ − LC experiments [4, 5].The top pair production is followed by a decay chain with intermediate gauge bo-son states, i.e. the full process under consideration is e + e − → t ∗ ¯ t ∗ → b ¯ bW + W − → b ¯ b f . The widths of both the top quark and the W -boson are large, and one nec-essarily needs to take into account corresponding Finite-Width Effects (FWE). Inthe framework of the standard perturbative approach, these effects are typicallydescribed by means of dressed propagators which are regularized by the total decaywidth. In order to analyze the full process of the top pair production relevant forphenomenological studies, we also have to take into account the background con-tribution coming from many other topologically different diagrams leading to thesame six-fermion final states, which is a rather non-trivial task.The Born-level cross-sections of the processes e + e − → b ¯ bu ¯ dµ − ¯ ν µ and e + e − → b ¯ b q were calculated in Refs. [6, 7] and [8], respectively. Other exclusive reactionswith b ¯ bd ¯ uµ + ν µ , b ¯ bc ¯ sd ¯ u and b ¯ bµ + ν µ τ − ¯ ν τ final states were considered in Ref. [5]. Inparticular, it was shown that the contribution of the top-pair signal e + e − → t ∗ ¯ t ∗ → b ¯ b f is dominant, but the background (caused by one-resonant or non-resonantdiagrams) can be quite significant too. However, it can be drastically decreased byapplying certain kinematical cuts on the appropriate invariant masses.The QCD corrections for the reaction e + e − → t ¯ t in the continuum abovethe threshold were previously obtained in Refs. [9, 10]. As well as the one-loopEW corrections were calculated in many papers (for corresponding references, seee.g. Introduction in Ref. [11]). Concerning radiative corrections (RC) to reaction e + e − → b ¯ b f with six-fermion final states, the situation is more complicated andless clear [11]. At the tree level, any of the reactions receives contributions from sev-eral hundreds of diagrams. The calculations of the full O ( α ) radiative correctionsare very complicated, and different approximation schemes are typically applied.The most detailed analysis of the exclusive reactions e + e − → b ¯ bµ + ν µ µ − ¯ ν µ and e + e − → b ¯ bd ¯ uµ − ¯ ν µ was performed in Ref. [11]. In this paper, the cross-sections werecalculated taking into account the leading radiative corrections, such as the initialstate radiation (ISR) and factorizable EW corrections to the on-shell top-pair pro-duction, to the decay of the top quark into bW and to the subsequent decays of the W -bosons. Usually, such calculations are carried out automatically by Monte Carlotechniques (see Ref. [11] and references therein).In this work, we consider reactions like e + e − → t ∗ ¯ t ∗ → b ¯ b f with any four-fermion final states 4 f . The analysis is performed in the framework of the smearedmass unstable particles model (below, SMUP model) [12, 13]. Due to exact factor-ization at intermediate t, ¯ t and W + , W − states, the cross-section can be representedin a simple analytical form which is convenient for analytical and numerical anal-eptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final Mass shell smearing effects in top pair production e + e − → t ∗ ¯ t ∗ → b ¯ bW + W − → b ¯ b f . ysis. So far, we have applied the SMUP approach only for unstable gauge bosonproduction and decay (see e.g. Refs. [14, 15, 16]). As a continuation of our earlierstudies, in this work we test the SMUP approach for the case of unstable fermions,i.e., specifically, top quarks. In our calculations, we take into account NLO radiativeEW and QCD factorizable corrections which dominate close to t ¯ t threshold. Also,we illustrate the influence of the mass smearing effects and various radiative correc-tions (RC’s) on the differential cross-sections. The results are compared with onescalculated by using the standard perturbative methods [11], where cross-sectionswere represented for case of full 2 → t ¯ t state, which were considered in detail in many previous studies(see, for instance, Ref. [17] and references therein). We postpone this issue for aforthcoming study.
2. The model cross-section of the top-pair production and decayat the tree level
The process of top-pair production with subsequent decay e + e − → t ∗ ¯ t ∗ → b ¯ bW + W − → b ¯ b f is schematically represented in Fig. 1. The full process containstwo steps with unstable intermediate time-like states, namely, t, ¯ t and W + , W − states. In this case, as was shown in Ref. [13], the double factorization takesplace and can be described in the framework of the SMUP model [12]. Due tothis factorisation, the full process can be divided into three stages: e + e − → t ∗ ¯ t ∗ , t ∗ ¯ t ∗ → b ¯ bW + W − and W + W − → f . Here, the top-quarks and W -bosons aretreated as unstable particles, and finite-width effects should be taken into account.The SMUP model cross-section of the first reaction e + e − → t ∗ ¯ t ∗ can be writteneptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final V. I. Kuksa, R. S. Pasechnik, D. E. Vlasenko as [12] σ ( e + e − → t ∗ ¯ t ∗ ) = Z sm Z ( √ s − m ) m σ ( e + e − → t ( m )¯ t ( m )) ρ t ( m ) ρ t ( m ) dm dm , (1)where m ≈ M b ( M b is the bottom quark mass) is the threshold value of the topmass variable, σ ( e + e − → t ( m )¯ t ( m )) is the cross-section of top pair productionwith random masses m and m and ρ t ( m ) is the probability density which describesthe mass smearing of top quarks. In our calculations we take it in the Lorentzianform as [12] ρ t ( m ) = 1 π m Γ t ( m )( m − M t ) + m Γ t ( m ) , (2)where Γ t ( m ) is the total decay width of the top quark with mass m . The decay mode t → bW has a very large branching ratio Br( t → bW ) ≈ . e + e − → t ∗ ¯ t ∗ → b ¯ bW + W − in the stable W -boson approximation. In order to take into account the instability of W -bosons wehave to express the top quark width Γ t ( m ) ≈ Γ( t → bW ) in Eq. (2) as a function ofsmeared W -boson mass Γ( t → bW ( M W )) with averaging over M W . Thus, the modelcross-section of the full inclusive process e + e − → t ∗ ¯ t ∗ → b ¯ bW + W − → b ¯ b P f f depicted in Fig. 1 has the following convolution form: σ ( e + e − → b ¯ b X f f ) = Z sm Z ( √ s − m ) m σ ( e + e − → t ( m )¯ t ( m )) × Z ( m − M b ) ( m − M b ) ρ t ( m , m W + ) ρ W ( m W + ) dm W + × (3) Z ( m − M b ) ( m − M b ) ρ t ( m , m W − ) ρ W ( m W − ) dm W − dm dm , where ρ W ( m ) is defined by Eq. (2). In order to describe an exclusive reaction e + e − → t ∗ ¯ t ∗ → b ¯ bf f f f we have to replace the total decay widths of W -bosons,which enter the numerator in Eq. (2), by corresponding exclusive ones (see Section4). The same result can be obtained exactly if one calculates the cross-section of thisprocess explicitly in the framework of the SMUP model by using dressed propaga-tors of unstable particles (UP’s). In Ref. [13] it was shown that exact factorizationof a decay chain process with UP’s in an intermediate state takes place when weexploit the model effective propagators for fermion and vector UP’s in the followingform ˆ D ( q ) = i ˆ q + qP F ( q ) , D µν ( q ) = − i g µν − q µ q ν /q P V ( q ) , (4)where P F ( q ) and P V ( q ) are the denominators of the fermion and vector bosondressed propagators, which contain corresponding total decay widths. The structureeptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final Mass shell smearing effects in top pair production e + e − → t ¯ t , e + e − → b ¯ bW + W − and e + e − → b ¯ b P f f . of numerators in Eq. (4) provides exact factorization and leads to a convolution-like expression (3) for the cross-section. So, there is a self-consistency between themodel and UP effective theory description of the processes with UP in intermediatestates. Thus, the process with a six-particles final state shown in Fig. 1 is describedby a simple analytical expression (3) with four integrations over smeared unstabletop and W -boson masses. Note, that the standard perturbative treatment of thesix-particle final states in general case leads to N = 3 · − W -bosons. Note, that the second case corresponds to the standard treatmentof the process e + e − → t ∗ ¯ t ∗ → b ¯ bW + W − in the stable W -boson approximation, andthe third case – to the full process shown in Fig. 1.From Fig. 2, one can see that the contribution of the top quarks’ FWE’s is signif-icant (up to a few percents in the near-threshold region), while the contribution of W -bosons’ FWEs is small. The comparison of our results with ones in the standardperturbative treatment shows that deviations are typically very small. For instance,it was obtained in Ref. [19], that σ ( e + e − → t ∗ ¯ t ∗ → b ¯ bW + W − ) for √ s = 500 GeVis equal to 629 fb for M t = 150 GeV and 553 fb for M t = 180 GeV. For the sameinput data, we have obtained 630 fb and 554 fb, respectively, which are in a goodagreement with the result mentioned above. This comparison proves the applicabil-ity of the SMUP model fermion propagator given by the first expression in Eq. (4).In Section 4, we make such a comparison for exclusive processes as well where bothSMUP model fermion and boson propagators Eq. (4) are used.It should be noticed also that we consider the FWE’s, which are significant inthe near-threshold region, but we do not include near-threshold effects caused byeptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final V. I. Kuksa, R. S. Pasechnik, D. E. Vlasenko possible intermediate t ¯ t bound states. Since the top mean lifetime is considerablyshorter than the hadronisation time, the bound state effect has no sharp resonantnature. However, it can be comparable with FWE’s or mass-smearing effects underconsideration, and this problem will be considered in more detail elsewhere.
3. Factorizable corrections to the cross-section
As it was shown in previous papers [11], the EW and QCD corrections give largecontributions to the cross-section of the top-pair production at energy scales closeto its threshold. In this Section, we describe the strategy of our model calcula-tions and give the total cross-section including the principal part of NLO EW andQCD corrections. Note, that the strategy of calculations and the choice of inputparameters are mainly caused and defined by the effective character of the modeltreatment. In the framework of the SMUP model, the instability (or finite width) ofunstable particles is accounted for by the smearing of their masses, i.e. by the prob-ability density function ρ ( m ). In turn, this function contains momentum dependentparameters M ( q ) and Γ( q ) in analogy with the standard perturbative treatmentwhich uses dressed propagators. So, in that sense the corrections of self-energy typeare already included at the “effective” tree level, and it is reasonable to use an ef-fective couplings, such as running coupling, absorbing the major part of vertex-typecorrections.In our calculations we have used the following input data [20]: α ( M Z ) = 0 . , α s ( M Z ) = 0 . , sin θ W ( M Z ) = ˆ s Z = 0 . ,M Z = 91 . , M W = 80 .
399 GeV , M t = 172 . . (5)The running coupling constants α k ( Q ) , k = 1 , , α k ( Q ) = α k ( M Z )1 − ( β k / π ) ln( Q /M Z ) , β k = (4 . , − / , − . (6)The cross-sections are calculated including the following corrections: • Vertex and self-energy type corrections for stable particles are mainly in-cluded into running couplings (6). • Self-energy corrections for unstable particles are included into the proba-bility density function ρ ( m ), which describes the smearing of UP’s masses. • Initial state radiation (ISR) is described by the photon radiation spectrum[21, 22], and the bremsstrahlung from the final t -quark states – by vertex Q -dependent factor [23]. • QCD corrections to the top production and decay are described by the ver-tex multiplicative factor [23].eptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final
Mass shell smearing effects in top pair production e + e − → b ¯ b P f f . • Contribution of the box diagrams to the total cross section was evaluatedat energy scales close to the threshold by using numerical FormCalc v7.3[24] routines.The higher order corrected cross-sections of the inclusive process e + e − → t ∗ ¯ t ∗ → b ¯ b P f f are shown in Fig. 3. There, the dotted line represents the Born modelcross-section, the dashed line – the cross-section with ISR and the solid line – thecross-section with total factorisable corrections (without box diagrams contribu-tion). From the figure, one can see that the main contribution is given by ISRcorrection, which significantly reduces the cross-section in the near-threshold en-ergy range and increases it at energy scales above ∼ √ s > .
4. The cross-sections of exclusive processes
So far, we have considered the cross-section of inclusive process e + e − → t ∗ ¯ t ∗ → b ¯ b P f f where the final state is summed up over all possible fermion flavors. Aswas noticed in the second Section, in order to get the cross-section of exclusiveprocess e + e − → t ∗ ¯ t ∗ → b ¯ bf f f f we can include the corresponding branchingratios Br( W → f f ) and Br( W → f f ). Acting this way we obtain σ ( e + e − → b ¯ bf f f f ) = σ ( e + e − → b ¯ b X f f )Br( W → f f )Br( W → f f ) . (7)eptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final V. I. Kuksa, R. S. Pasechnik, D. E. Vlasenko
Fig. 4. Invariant top mass distribution. where we omit intermediate virtual t ∗ ¯ t ∗ state for simplicity. This relation directlyfollows from the Eq. (3) when one substitutes a partial decay width of the W -boson into numerator of the probability distribution function ρ W ( m ) instead of thetotal width. It can be also derived by straightforward calculation of the σ ( e + e − → b ¯ bf f f f ) in the framework of the effective theory (see Ref. [13]). Fig. 5. The angular differential cross-sections for the process e + e − → b ¯ b P f f . The expressions for the branchings ratios Br( W → f f ) were considered indetail in Ref. [23]. Here, we use very simple but sufficiently precise formulae whichincorporate QCD corrections:Br( W → l ¯ ν l ) = 19(1 + 2 α s ( M Z ) / π ) , Br( W → u i ¯ d k ) = | V ik | (1 + α s ( M Z ) /π )3(1 + 2 α s ( M Z ) / π ) , (8)eptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final Mass shell smearing effects in top pair production where V ik are elements of the Cabibbo-Kobayashi-Maskawa mixing matrix. We,also, employ the QCD corrected expression for the top quark width [23, 25]:Γ( t → bW ) = 116 α ( M t ) | V tb | η QCD M t f ( M t , M W , M b ) , (9)where f ( M t , M W , M b ) = λ ( M b , M W ; M t ) (cid:18) ( M t − M b ) M t M W + M t + M b − M W M t (cid:19) ;(10) λ ( M b , M W ; M t ) = (cid:18) − M b + M W M t + ( M W − M b ) M t (cid:19) / ; η QCD = 1 − α s ( M t )3 π (cid:18) π − (cid:19) . Using Eqs. (7)–(9) we can calculate the exclusive cross-section for an arbitrarysix-fermion final state ( b ¯ bf f f f ). Such calculations taking into account the factor-izable EW corrections were performed within the standard perturbative approachfor the case of ( b ¯ bµ + ν µ µ − ¯ ν µ ) and ( b ¯ bµ + ν µ d ¯ u ) final states in Ref. [11]. In this work,the full set of topologically different Born diagrams leading to the same six-fermionfinal state was considered. It was shown, that certain cuts on invariant masses ofthe W b and f i f k pairs, which correspond to intermediate t, ¯ t and W + , W − statesfor signal diagrams, significantly reduce the relative contribution of the background(see Table 1).In Tables 1 and 2 the cross-sections are given for two distinct reactions(1) : e + e − → b ¯ bµ + ν µ µ − ¯ ν µ , (2) : e + e − → b ¯ bµ + ν µ d ¯ u. (11)for the energies √ s = 430 , , , GeV. In Table 1 the cross-sections are pre-sented in the Born approximation for total set of diagrams ( σ ( k ) Born (total)) and for thesignal diagrams ( σ ( k ) Born ( t ∗ ¯ t ∗ )), where k = 1 , δ i < .
1, where δ i is the deviation ofthe ratio m invi /M i from unity and index i is related to different t, ¯ t, W + , W − states(for more details, see Ref. [11]). Table 1. Born-level cross-sections of the processes (1) and (2) in Eq. (11). √ s , GeV σ (1) Born (total) σ (1) Born ( t ∗ ¯ t ∗ ) σ (2) Born (total) σ (2) Born ( t ∗ ¯ t ∗ )430 5.9117 5.8642 17.727 17.592500 5.3094 5.2849 15.950 15.8551000 1.6387 1.6369 4.9134 4.9106 In Table 2 the results for the total cross sections (in fb) of processes (1) and(2) from Eq. (11) in the Born approximation are shown in the second column.The cross-sections with separate ISR and factorizable EW (FEWC) corrections arepresented in the third and forth columns, respectively, and the cross-section witheptember 20, 2018 21:24 WSPC/INSTRUCTION FILE Toppair˙final V. I. Kuksa, R. S. Pasechnik, D. E. Vlasenko both the FEWC and ISR corrections included – in the fifth column. All values arecalculated with the kinematical cuts mentioned above.
Table 2. Comparison of the exclusive cross-sections of Ref. [11] and ones obtained in the presentwork. √ s, GeV σ t ∗ ¯ t ∗ Born σ Born+ISR σ Born+FEWC σ Born+ISR+FEWC e + e − → bν µ µ + ¯ bµ − ¯ ν µ , Ref. [11]430 5 . . . . . . . . . . . . e + e − → bν µ µ + ¯ bµ − ¯ ν µ , this work430 5 . . . . . . . . . . . . e + e − → bν µ µ + ¯ bd ¯ u , Ref. [11]430 17 . . . . . . . . . . . . e + e − → bν µ µ + ¯ bd ¯ u , this work430 17 . . . . . . . . . . . . From Table 2, it follows that the differences of the model and standard Borncross-sections are of an order of 0.1 percent and ISR correction increases it onlyslightly. In principle, these deviations can be further reduced. The situation becomesworse, when we take into account all major corrections. The deviations increaseand become up to a few percents. This discrepancy is caused by the fact that inRef. [11] an additional contribution from the non-signal (background) diagramswas included while we consider the signal contribution only. Moreover, we do notinclude the contribution of the box diagrams which becomes very important atlarge energies far from the threshold. According to estimations in the frameworkof the standard perturbative treatment, the box diagrams contribution is of anorder of a few percents in the near-threshold energy range. Rough estimations inthe framework of the SMUP model give the box contribution equal to 1 . − Mass shell smearing effects in top pair production
5. Conclusion
The production of the t ¯ t pair and its subsequent decay into six fermion final statesin e + e − annihilation has been previously analyzed within the standard perturbativetreatment in a vast literature. In this work, we performed the corresponding analysisin the framework of SMUP model. So far, this approach was applied mainly to thegauge boson production, where the structure of the model boson propagators wassuccessfully tested [14, 15, 16]. In the present work, we have tested the structureof the model fermion propagator, and the top quark production mechanism hasbeen chosen as an important example. It was shown that the results of Born-levelcalculations are in a good agreement with the standard perturbative ones, providingthe applicability of the SMUP approach to the top-quark production and decayprocesses.The SMUP model provides simple analytical expressions for the total cross-sections of inclusive and exclusive processes with top quark pair production and itssubsequent decay. It is a convenient and simple instrument for description of com-plicated multi-step processes with unstable particles participation. The precision ofthis approach at the tree level is of an order of 0.1 percent or better. The methodgives a possibility to include, in principle, all factorizable corrections. Our approachcan be useful in a preliminary analysis of complicated processes with intermediatetime-like top quark exchanges within the Standard Model and beyond. References
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